The bright (V=3.86) star $\beta$ Pictoris is a nearby young star with a debris disk and gas giant exoplanet, β Pictoris b, in a multi-decade orbit around it. Both the planet's orbit and disk are almost edge-on to our line of sight. We carry out a search for any transiting planets in the β Pictoris system with orbits of less than 30 days that are coplanar with the planet β Pictoris b. We search for a planetary transit using data from the BRITE-Constellation nanosatellite BRITE-Heweliusz, analyzing the photometry using the Box-Fitting Least Squares Algorithm (BLS). The sensitivity of the method is verified by injection of artificial planetary transit signals using the Bad-Ass Transit Model cAlculatioN (BATMAN) code. No planet was found in the BRITE-Constellation data set. We rule out planets larger than 0.6 RJ for periods of less than 5 days, larger than 0.75 $\mathrm{R_J}$ for periods of less than 10 days, and larger than 1.05 RJ for periods of less than 20 days.